Photographic (or other optically captured images) images may be blurred by relative motion between an imaging system or camera and an object of interest. An example of a motion-blurred image is illustrated in FIG. 1. Motion-blurred images commonly occur when the imaging system is in moving vehicles or held by a human hand. OCR of blurred text of blurred text may be inaccurate or in some cases may be impossible.
The methods used to restore such images typically involve convolving the real image with a point spread function (PSF):
                                          g            ⁡                          (                              x                ,                y                            )                                =                                                    ∑                                  k                  =                  1                                K                            ⁢                                                ∑                                      m                    =                    1                                    M                                ⁢                                                      f                    ⁡                                          (                                              k                        ,                        m                                            )                                                        ⁢                                      h                    ⁡                                          (                                                                        x                          -                          k                                                ,                                                  y                          -                          m                                                                    )                                                                                            +                          n              ⁡                              (                                  x                  ,                  y                                )                                                    ,                            (        1        )            where    f(x,y) is the non-blurred image;    g(x,y) is the image obtained from the camera;    h(x,y) is the PSF;    n(x,y) is the noise; and    K and M is the size of the image in pixels.
In the case of a blurred image, the PSF is a function of one argument, i.e. h(x,y)=h1D(x sin(α)+y cos(α)), where α is the angle of the blur.
In the Fourier space, the equation (1) becomes:
                                                        G              ⁡                              (                                  p                  ,                  q                                )                                      =                                                            F                  ⁡                                      (                                          p                      ,                      q                                        )                                                  ⁢                                  H                  ⁡                                      (                                          p                      ,                      q                                        )                                                              +                              N                ⁡                                  (                                      p                    ,                    q                                    )                                                              ,                                          ⁢          where          ,                      for            ⁢                                                  ⁢            example                    ,                                          ⁢                                    G              ⁡                              (                                  p                  ,                  q                                )                                      =                                          ∑                                  x                  =                  1                                K                            ⁢                                                ∑                                      y                    =                    1                                    M                                ⁢                                                      ⅇ                                                                                                                        2                            ⁢                                                                                                                  ⁢                            π                                                    K                                                ⁢                        ⅈ                        ⁢                                                                                                  ⁢                        px                                            +                                                                                                    2                            ⁢                                                                                                                  ⁢                            π                                                    M                                                ⁢                        ⅈ                        ⁢                                                                                                  ⁢                        qy                                                                              ⁢                                      g                    ⁡                                          (                                              x                        ,                        y                                            )                                                        ⁢                                                                          ⁢                  and                                                                    ⁢                                  ⁢                              g            ⁡                          (                              x                ,                y                            )                                =                                    1                              K                ·                M                                      ⁢                                          ∑                                  p                  =                  1                                K                            ⁢                                                ∑                                      q                    =                    1                                    M                                ⁢                                                      ⅇ                                          -                                              (                                                                                                                                            2                                ⁢                                                                                                                                  ⁢                                π                                                            K                                                        ⁢                            ⅈ                            ⁢                                                                                                                  ⁢                            px                                                    +                                                                                                                    2                                ⁢                                                                                                                                  ⁢                                π                                                            M                                                        ⁢                            ⅈ                            ⁢                                                                                                                  ⁢                            qy                                                                          )                                                                              ⁢                                                            G                      ⁡                                              (                                                  p                          ,                          q                                                )                                                              .                                                                                                          (        2        )            
Since the function G(p,q) and the others are periodical G(p,q)=G(p−K, q)=G(p,q−M), it is assumed everywhere below that the p and q variables may have either positive or negative values.
H(p,q) is the Fourier transform of the PSF, often called optical transfer function (OTF). In the case of a blurred image, the OTF is a complex function of one argument H(p,q)=H1D(p·sin(α+π/2)+q·cos(α+π/2)).
Also, the Wiener filter may be used to restore images:
                                                        F              ^                        ⁡                          (                              p                ,                q                            )                                =                                                                      H                  -                                ⁡                                  (                                      p                    ,                    q                                    )                                                                                                                                            H                      ⁡                                              (                                                  p                          ,                          q                                                )                                                                                                  2                                +                                                                                                                          N                        ⁡                                                  (                                                      p                            ,                            q                                                    )                                                                                                            2                                                                                                                            F                        ⁡                                                  (                                                      p                            ,                            q                                                    )                                                                                                            2                                                                        ⁢                          G              ⁡                              (                                  p                  ,                  q                                )                                                    ,                            (        3        )            where    {circumflex over (F)}(p,q) is the estimated function (which hopefully is close to F(p,q)) and    H−(p,q) is the complex conjugate of H(p,q).
This filter minimizes the root mean square deviation of the restored image from the real image
            ∑              x        ,        y                                  ⁢                  (                                            f              ^                        ⁡                          (                              x                ,                y                            )                                -                      f            ⁡                          (                              x                ,                y                            )                                      )            2        ,provided that the average noise value is 0.
Therefore, in order to restore a blurred image, one needs to know:                1. the OTF H(p,q), and        2. the signal-to-noise ratio in the impulse space        
                                      N          ⁡                      (                          p              ,              q                        )                                      2                                        F          ⁡                      (                          p              ,              q                        )                                      2        .
U.S. Pat. No. 6,470,097 Oct. 22, 2002 describes an iteration method for finding a non-blurred image. At each step of this method, total variation regularization is performed to minimize the energy function with the image blur. The type of distortion is set as a parameter and is predefined. The length of the blur and its direction are also predefined. Additionally, the image is restored from a sequence of images rather than from one blurred image.
U.S. Pat. No. 6,859,564 Feb. 22, 2005 describes a method where the OTF is determined from the scaled αth power of the smoothed magnitude of the blurred image and noise.
Other methods for restoring blurred images are described, for example, in the review by D. Kundur and D. Hatzinakos, “Blind Image Deconvolution Revisited,” IEEE Signal Processing Magazine, vol. 13, no. 6, pp. 61-63, November 1996, and in other sources.